Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
752 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C7×#390 | C2 |
| prime radical | C2×#6503 | C2 |
| hypercenter | #390 | D7 |
| hypocenter | C7 | #2858 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#6451 | C2 | semidirect |
| 3. C2×#6448 | C2 | semidirect |
| 4. C7×#390 | C2 | semidirect |
| 5. #89993 | C2 | semidirect |
| 6. #89760 | C2 | semidirect |
| 7. #89729 | C2 | semidirect |
| 8. #89711 | C2 | semidirect |
| 9. C42×Dic7 | C2 | semidirect |
| 63. C4×Dic7 | D4 | semidirect |
| 71. #390 | D7 | semidirect |
| 143. C4×C7 | #124 | semidirect |
| 146. Dic7 | #124 | semidirect |
| 221. C7 | #2858 | semidirect |
| 243. C4 | #6448 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6503 | 2 |
| C2×#6451 | 2 |
| C2×#6448 | 2 |
| C7×#390 | 2 |
| #89993 | 2 |
| #89760 | 2 |
| #89729 | 2 |
| #89711 | 2 |
| C42×Dic7 | 2 |
| #2858 | 7 |